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Properties

Label 242.2.c.f
Level $242$
Weight $2$
Character orbit 242.c
Analytic conductor $1.932$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [242,2,Mod(3,242)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("242.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 242.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.93237972891\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{4} - \beta_{2} - 1) q^{2} + ( - \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{3} + \beta_{6} q^{4} - \beta_{7} q^{5} + (\beta_{7} - \beta_{2}) q^{6} + (3 \beta_{6} + \beta_1) q^{7}+ \cdots + (6 \beta_{5} + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} - 6 q^{7} - 2 q^{8} - 2 q^{9} - 8 q^{12} - 6 q^{13} - 6 q^{14} + 6 q^{15} - 2 q^{16} - 2 q^{18} - 6 q^{19} - 24 q^{23} + 2 q^{24} + 4 q^{25} - 6 q^{26} + 8 q^{27}+ \cdots + 40 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\) \(123\)
\(\chi(n)\) \(\beta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−0.535233 + 1.64728i
0.535233 − 1.64728i
−1.40126 − 1.01807i
1.40126 + 1.01807i
−1.40126 + 1.01807i
1.40126 − 1.01807i
−0.535233 − 1.64728i
0.535233 + 1.64728i
−0.809017 + 0.587785i −0.844250 − 2.59833i 0.309017 − 0.951057i −1.40126 − 1.01807i 2.21028 + 1.60586i 0.391818 − 1.20589i 0.309017 + 0.951057i −3.61153 + 2.62393i 1.73205
3.2 −0.809017 + 0.587785i 0.226216 + 0.696222i 0.309017 − 0.951057i 1.40126 + 1.01807i −0.592242 − 0.430289i 1.46228 − 4.50045i 0.309017 + 0.951057i 1.99350 − 1.44836i −1.73205
9.1 0.309017 − 0.951057i −0.592242 + 0.430289i −0.809017 − 0.587785i −0.535233 − 1.64728i 0.226216 + 0.696222i −3.82831 − 2.78143i −0.809017 + 0.587785i −0.761449 + 2.34350i −1.73205
9.2 0.309017 − 0.951057i 2.21028 − 1.60586i −0.809017 − 0.587785i 0.535233 + 1.64728i −0.844250 − 2.59833i −1.02579 − 0.745282i −0.809017 + 0.587785i 1.37948 − 4.24561i 1.73205
27.1 0.309017 + 0.951057i −0.592242 − 0.430289i −0.809017 + 0.587785i −0.535233 + 1.64728i 0.226216 − 0.696222i −3.82831 + 2.78143i −0.809017 − 0.587785i −0.761449 − 2.34350i −1.73205
27.2 0.309017 + 0.951057i 2.21028 + 1.60586i −0.809017 + 0.587785i 0.535233 − 1.64728i −0.844250 + 2.59833i −1.02579 + 0.745282i −0.809017 − 0.587785i 1.37948 + 4.24561i 1.73205
81.1 −0.809017 − 0.587785i −0.844250 + 2.59833i 0.309017 + 0.951057i −1.40126 + 1.01807i 2.21028 − 1.60586i 0.391818 + 1.20589i 0.309017 − 0.951057i −3.61153 − 2.62393i 1.73205
81.2 −0.809017 − 0.587785i 0.226216 − 0.696222i 0.309017 + 0.951057i 1.40126 − 1.01807i −0.592242 + 0.430289i 1.46228 + 4.50045i 0.309017 − 0.951057i 1.99350 + 1.44836i −1.73205
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.2.c.f 8
11.b odd 2 1 242.2.c.g 8
11.c even 5 1 242.2.a.e yes 2
11.c even 5 3 inner 242.2.c.f 8
11.d odd 10 1 242.2.a.c ✓ 2
11.d odd 10 3 242.2.c.g 8
33.f even 10 1 2178.2.a.y 2
33.h odd 10 1 2178.2.a.s 2
44.g even 10 1 1936.2.a.y 2
44.h odd 10 1 1936.2.a.v 2
55.h odd 10 1 6050.2.a.cv 2
55.j even 10 1 6050.2.a.cc 2
88.k even 10 1 7744.2.a.bt 2
88.l odd 10 1 7744.2.a.bq 2
88.o even 10 1 7744.2.a.cv 2
88.p odd 10 1 7744.2.a.cs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
242.2.a.c ✓ 2 11.d odd 10 1
242.2.a.e yes 2 11.c even 5 1
242.2.c.f 8 1.a even 1 1 trivial
242.2.c.f 8 11.c even 5 3 inner
242.2.c.g 8 11.b odd 2 1
242.2.c.g 8 11.d odd 10 3
1936.2.a.v 2 44.h odd 10 1
1936.2.a.y 2 44.g even 10 1
2178.2.a.s 2 33.h odd 10 1
2178.2.a.y 2 33.f even 10 1
6050.2.a.cc 2 55.j even 10 1
6050.2.a.cv 2 55.h odd 10 1
7744.2.a.bq 2 88.l odd 10 1
7744.2.a.bt 2 88.k even 10 1
7744.2.a.cs 2 88.p odd 10 1
7744.2.a.cv 2 88.o even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 6T_{3}^{6} - 16T_{3}^{5} + 44T_{3}^{4} + 32T_{3}^{3} + 24T_{3}^{2} + 16T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 6T_{7}^{7} + 30T_{7}^{6} + 144T_{7}^{5} + 684T_{7}^{4} + 864T_{7}^{3} + 1080T_{7}^{2} + 1296T_{7} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 27 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 6)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 279841 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} + \cdots + 104976 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$59$ \( T^{8} + 12 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T - 2)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} - 12 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$73$ \( T^{8} + 12 T^{7} + \cdots + 20736 \) Copy content Toggle raw display
$79$ \( T^{8} - 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$83$ \( T^{8} + 6 T^{7} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( (T^{2} - 6 T - 3)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
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